Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

Benâ€™s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

Got It game for an adult and child. How can you play so that you know you will always win?

Here are two kinds of spirals for you to explore. What do you notice?

Can you explain the strategy for winning this game with any target?

Are these statements relating to odd and even numbers always true, sometimes true or never true?

This challenge encourages you to explore dividing a three-digit number by a single-digit number.

This task follows on from Build it Up and takes the ideas into three dimensions!

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

We can arrange dots in a similar way to the 5 on a dice and they usually sit quite well into a rectangular shape. How many altogether in this 3 by 5? What happens for other sizes?

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

An investigation that gives you the opportunity to make and justify predictions.

Try adding together the dates of all the days in one week. Now multiply the first date by 7 and add 21. Can you explain what happens?

This activity involves rounding four-digit numbers to the nearest thousand.

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

This challenge asks you to imagine a snake coiling on itself.

Do you notice anything about the solutions when you add and/or subtract consecutive negative numbers?

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

Are these statements always true, sometimes true or never true?

Can you find all the ways to get 15 at the top of this triangle of numbers? Many opportunities to work in different ways.

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

Tom and Ben visited Numberland. Use the maps to work out the number of points each of their routes scores.

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

How many centimetres of rope will I need to make another mat just like the one I have here?

Delight your friends with this cunning trick! Can you explain how it works?

Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

Strike it Out game for an adult and child. Can you stop your partner from being able to go?

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

Find the sum of all three-digit numbers each of whose digits is odd.

What happens when you round these numbers to the nearest whole number?

This task encourages you to investigate the number of edging pieces and panes in different sized windows.